On a numerical method to solve contact problems
Citation for published version (APA):Baaijens, F. P. T. (1987). On a numerical method to solve contact problems. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR258682
DOI:
10.6100/IR258682
Document status and date: Published: 01/01/1987 Document Version:
Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication:
• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website.
• The final author version and the galley proof are versions of the publication after peer review.
• The final published version features the final layout of the paper including the volume, issue and page numbers.
Link to publication
General rights
Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain
• You may freely distribute the URL identifying the publication in the public portal.
If the publication is distributed under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license above, please follow below link for the End User Agreement:
www.tue.nl/taverne Take down policy
If you believe that this document breaches copyright please contact us at: openaccess@tue.nl
providing details and we will investigate your claim.
CONTACT PROBLEMS
CONTACT PROBLEMS
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICU~, PROF. DR. F. N. HOOGE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN
DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP DINSDAG 20 JANUARI 1987 TE 16.00 UUR
DOOR
FRANCISCUS PETRUS THOMAS BAAIJENS
Prof. dr. ir. J. D. lanssen en
Prof. dr. ir. D. H. van Campen
Abstract. Introduction. 2 Contact theory.
2.1 Introduction.
2.2 Some kinematic and dynamic quantities. 2.3 Contact conditions.
2.4 Statement of the problem.
2.5 A weak tormulation of the contact problem of two deformable bodies.
2.6 The Lagrange multiplier method in case of frictionless finite elasticity contact problems.
2.7 Other methods.
3 A constitutive equation for a class of frictional phenomena. 3.1 Introduction.
3.2 Some kinematic quantities.
3.3 Objective rates of the shear stress vector. 3.4 Reversible and irreversible behaviour. 3.5 Slip function.
3.6 A constitutive equation for irreversible slip. 3.7 Two-dimensional problems.
4 Discretization.
4.1 Introduction. 4.2 Approximations.
4.3 Discretized formulation. 4.4 Discrete contact conditions. 4.5 Non-smooth boundaries. 4.6 An example.
5.2 A set of non-linear (vector) equations. 5.3 The solution strateqy.
5.4 Shift step.
6 Examples.
6.1 Introduction.
6.2 An upsettinq example. 6.3 Effect of gridsize.
6.4 A 'corner contact' situation. 6.5 Two deformable bodies.
7 Discussion.
References.
Appendices.
A The Gateaux derivative of the potential energy functional. B The Gateaux derivatives of q.
c
The two-dimensional gradient operator.D The relation between the BA and the BB derivative. E Calculation of the frictional stress.
Samenvatting.
In many forming processes contact phenomena play an important rele. The analysis of such processes is hampered on the one hand by the
(usually large) relative displacements between the contact bodies which lead to difficulties in modelling the contact constraints, and on the ether hand by the available, simple constitutive equations for frictional phenomena which lead to unsatisfactory results. In this thesis a salution to these problems is proposed.
In chapter 2 a general tormulation of the contact constraints is given which is valid for each magnitude of the relative
displacements. Based upon the methad of weighted residuals a weak tormulation is derived that incorporates these constraints.
An important part of this thesis is formed by the tormulation of a constitutive equation with which a large class of frictional phenomena can be described in a flexible way. This model shows a streng analogy with existing models for elasto-plastic material behaviour. Distinction is made between reversible (micro-) slip and irreversible slip. The amount of irreversible slip is determined with the help of a slip criterion, while the direction fellows from a quasi-associated slip rule.
Based upon the finite element methad numerical simulations are made. In the discretization of the governing equations difficulties arise with respect to the contact constraints. A salution is
described in chapter 4.
The resulting system of non-linear (vector) equations is solved iteratively by application of Newton's method. As a consequence of the contact phenomena (including the frictional behaviour) these equations have a special structure which can be solved effectively with a partitioned salution process.
Same characteristics of the developed methad are illustrated in some problems in chapter 6. This thesis is closed with a critica! discussion of a number of choices and assumptions that (necessarily) had to be made at a number of places in this thesis.
Chapter 1 Introduction.
During the last decade, the finite element method has gained wide application in the analysis of elasto-plastic deformation prablems. However, applications to the simulation of practical metal forming .processes are limited. One of the reasans for this is that only rather simple geometrical contact constraints can be introduced. A similar remark holds with respect to the friction models that are available. Nevertheless, the simulations show that the proper defini-tion and handling of the contact constraints ,and the correct choice of the friction model, play a vital role in the analysis. Within this perspective, the aim of this thesis is to formulate the geometrical contact constraints for finite deformation problems, to introduce a more general friction model and to propose a numerical methad such that an enlarged class of metal forming processes can be simulated.
From a physical point of view, the contact constraints are really quite simple, since contact arises because no material point of one body can occupy the same position as a material point of another body. The difficulty however is to describe this
im-penetrability in a mathematica! sense. For small deformation/rotation problems with smoothly bounded bodies, it is well known how this can be done (Gladwell, 1980). For finite deformation problems, however, there is no generally accepted method. One of the main aspects of this thesis is the introduetion of a fairly general tormulation of impenetrability which is quite suitable for numerical purposes.
It is widely recognized that many metal forming processes are largely dominated by the frictional behaviour between the contact bodies. Therefore it is, at first sight, rather surprising that there is a lack of realistic models for the actual frictional phenomena taking place under metal forming conditions. This, most probably, is caused by the fact that experiments in this area are extremely dif-ficult because the accurate direct measurement of the contact stresses is almast impossible. This causes the use of 'global' fric-tion models such as eaulomb's model or the so-called constant
friction factor model. In many situations both models lead to rather discutable results. With the use of more refined experimental tech-niques (Starmans, 1986), more sophisticated friction models should be developed. A fairly general approach was given by Fredricsson (1976)
who noticed that friction essentially shows a behaviour analogous to metal plasticity in the sense that only if a limiting shear stress, which might be a tunetion of many parameters, is reached, substantial irreversible slip occurs. Later Pires (1982) introduced, inspired by the workof Johnson (1955) and Courtney-Pratt and Eisner (1957), the concept of reversible micro-slip. In this thesis these two ideas are mingled for finite deformation problems. Ideas similar to those in elasto-plastic material models, such as the use of objective rates and a quasi-associated sliprule, are introduced. The proposed model provides a starting point for further theoretica! and experimental investigations in this area.
In general the considered class of problems is described by a system of highly nonlinear equations and inequalities. Analytica! solutions are hard or impossible to obtain and a common alternative to solve these kind of problems is to apply the finite element method. Yet, to do so a weak tormulation is required. The difficulty bere is the inclusion of the contact constraints because they are written in an inequality farm. For infinitesimal deformation problems it is common practice to apply either the Lagrange multiplier method, the penalty tunetion metbod or the augmented Lagrange method.
Formally, neither of these methods can be applied because the con-sidered problems cannot be written in a minimization farm. Therefore the metbod of weighted residuals is applied in this thesis, where attention is focussed on the incorporation of the contact
constraints. It is shown that the proposed methad is closely related to the Lagrange multiplier method.
When the finite element methad is used, some difficulties arise with respect to the incorporation of the contact constraints in the discretized sense. If the integration of the contact stresses and the contact constraints along a candidate contact area is carried out improperly, some rather unexpected results occur. This nuisance can be evereome by using a so-called selective reduced integration technique. This technique has been studied extensively for linear elastic contact problems by Oden and Kikuchi (1980). The impact of this technique to the handling of the contact constraints is dis-cussed in detail.
The numerical salution of the discretized tormulation is to some extend standard. Yet, some new features are introduced. Since
the main scope of this thesis is to introduce a numerical methad to solve contact problems in finite deformation mechanics, some of the possibilities of the proposed methad are illustrated for a number of, rather academie, examples.
Chapter 2 Contact theory.
2.1 Introduction.
The scope of this chapter is to give a weak tormulation of a fairly large class of contact problems in such a manner that finite element techniques can be applied to generate numerically approximate solu-tions. Specifically, time independent, physical and geometrical non-linear contact problems are considered. Typically, the theory will be applied to finite strain elasto-plastic deformation problems.
Before a weak tormulation can be derived, the impenetrability of the contact bodies must be described mathematically. It is well-known how this can be done if the displacements and rotations are sufficiently small and the contact surfaces are smooth enough. How-ever, if this is not the case, little is known about how impenetrabi-lity should be formulated mathematically and how it should be taken into account. In 2.3 a new approach is introduced to describe the impenetrability for nonlinear problems, which is quite suitable for numerical purposes.
Another important aspect of contact is the interaction between the contact bodies. In this thesis it is assumed that no adhesion may occur in the direction perpendicular to the contact surface and it is assumed that the shear stresses can be determined with the help of an, as yet unspecified, friction model. Probably the most well-known example of such a model is Coulomb's law. In chapter 3 a somewhat more sophisticated model is presented.
As in linear contact problems, the contact conditions are writ-ten in an inequality form, at least as far as the geometrical con-straints are concerned. Due to the nonlinearity of the considered problems, classica! optimization techniques (such as the Lagrange multiplier rule or the penalty tunetion method) cannot be applied to take care of the inequality boundary conditions. Therefore an exten-sion of the, more generally applicable, weighted residuals method is discussed in 2.5.
For some finite elasticity problems the Lagrange multiplier rule can be applied. It is shown in 2.6 that this method can be seen as a special case of the weighted residuals method as discussed in
2.5. It is worthwile to study this approach because for the discre-tized situation some existence and uniqueness results are known.
Finally, some comments are made with respect to the applicabi-lity of two ether methods being the penalty function method and the augmented Lagrange method.
2.2 Some kinematic and dynamic quantities.
Throughout this thesis the suffix a denotes quantities with respect to body 1 where a = A or a = B. Consider a three dimensional body
a
L , that accupies a bounded open domain Q (t) with boundary r (t) at
a a a
the current state t. Let
Q
(t) denote the closure of Q (t), i.e.a a
Q
(t) = Q (t)ur (t). The boundary r (t) is assumed sufficientlya a a a
smooth such that at each point of r (t) the unit outward normal a ~ a can be defined.
In
Q
(t) a material reference system V is introduced, sucha a
that each material point of Q (t) is identified uniquely by three a
independent material coordinates r1a, r2a and r3a, beinq the compo-nents of a column r . The columns r of all material points of Q (t)
~a ~a a
are the elements of an invariable set R . The material point of
Q
(t)a a
with V -coordinates r is denoted by P (r ).
a ~a a ~a
A material point of r (t) can be identified both by r and by
a ~a
two material coordinates m1a and m2a, taken as the components of a column m . The columns m constitute a material reference system B
-a. ~a. a.
in r (t). The columns m of all material points of r (t) are the
ele-a ~a. a.
ments of an invariable set M . A material point of r (t) with B -co-a. a. a. ordinates m is denoted by P (m ).
~a. a. ~a.
The current position vector
x
of P (r ) is a function of ra a ~a. ~a
and t, and this function
x
a. :R a.•Q
a. (t) is assumed to be invertible, continuous and sufficiently differentiable. The entire field of posi-tien veetorsx
(r ,t) defines the shape of body L and is called thea. ~a. • a.
-contiguration field of
Q
(t). The Jacobian of x :R •Q (t) isdeno-a a a. a.
ted by J (r ,t). Similar remarks hold for the tunetion
x
:M •r (t)a. ~a • a a a
with Jacobian J (m ,t), where
x
(m ,t) is the current position vectora. ~a a ~a
of point P (m ) .
In the preceding,
Q
(t) and r (t) are considered as the set ofa a
end points of the position veetors of the material points of Q (t) or
a
r (t). Sometimes it is more convenient to consider
Q
(t) and r (t) asa a a
the set of position veetors belonging to the material points of
Q (t)
a..
-
..
or r (t), i.e. x €Q (tl or x er (t).
a a a a a
The contiguration at time t = t0 is supposed to be known and is called the reference configuration. A quantity measured at this con-tiguration is supplied with suffix 0, e.g. the contiguration field of
-
..
..
Q~(t
0
) is written as x 0(r l =x (r ,t0).
.... a ~a a ~a
The gradient operators with respect to the current and to the
..
..
reference contiguration are denoted by V and
v
0, respectively. The deformation at material point P (r ) in state t as compared to statea ~a .. .. c
t
0 is characterized by the deformation tensor F a (r ,t) = ~a (V0x ) . It a
can be shown that the volumes dQ (t) and dQ
0 of an infinitesimal
a a
small material element in the current and in the reference configura-tion, respectively, are related by
dQ
6 = det(F )
a a =_a_= dQaO (2.2.1)
where 6 is the volume change factor. Likewise, the surface change
a factor is given by 6 a dr J = _a_ - _.!L. drao- jao (2. 2 .2)
Let Q be the symmetrical Cauchy stress tensor. The current
a
..
..
..
stress vector p
=
p (m ,t) on r (t) in P (m ) is equal to Q •n .a a ~a a a ~a a a
This vector is decomposed in a normal component, the normal stress vector ~na' and a tangential component, the shear stress vector ~ta' where
..
0 na o n · na..
a' o na..
P a •n..
a..
..
..
([-n n l•P a a a (2. 2. 3) (2.2.4)2.3 Contact conditions. The bodies LA and L
8 are assumed to be impenetrable. They may con-tact each other. A material point PA(~A) of rA(t) and a material point P
8
(~8
) of r8(t) are called current contact points if at state t
their positions coincide. The current contact surface rc(t) is the intersection of rA(t) and r
8(t). Single point contact is not
conside-red. The physically rough surfaces of the bodies are modelled by suf-ficiently smooth surfaces. These smooth surfaces may be seperated by a lubricant film, which is assumed to be sufficiently thin such that there is no need to consider it as far as the contact constraints are concerned. Since the boundaries are supposed to be smooth, the
nor--+ -+
mals nA(~A,t) and n
8
(~8
,t) are opposite at current contact points, i.e.( 2. 3. 1)
At current contact points PA(~A) and P
8
(~8
) the law of action and reaction applies, i.e.(2.3.2)
Because of (2.3.1) this implies that
( 2. 3. 3)
Hence the normal contact stresses onA and onB are equal. This
justi-ties the introduetion of the contact pressure on as
(2. 3. 4)
Adhesion is not allowed at current contact points, therefore the con-tact pressure must be less than or equal to zero
on i
o
on re (2.3.5)-+
The shear stress ota is determined with the help of a given constitu-tive equation which is discussed in detail later.
An important aspect of the description of the contact condi-tions is the mathematica! tormulation of the impenetrability of bath bodies. For that purposesome definitions are given. Let rCA(t) be a candidate contact area of rA(t), i.e. let rCA(t) contain all points of rA(t) that may contact r8(t), e.g. rc(t)crCA(t). The part rc
8(t) is defined likewise, i.e. fC(t)crCB(t). Then impenetrability is guaranteed if no point of rCA(t) penetrates o
8(t). In this case,
ar-bitrarily, body LA has been taken as the reierenee body and is called the contactor, while body L
8 is· called the target. Of course the roles of the bodies can be interchanged without affecting the results. In general this does not hold in the discretized case, as will be shown later.
Firstly, the classica! tormulation for small displacements, small rotations and regular boundaries is discussed. In that case the impenetrability condition is simple. At each point PA(m ) of r (t
0)
~A CA ~
the initia! gap s
0
(~A) between PA(~A) and r8(t0) measured along nAO can be evaluated, see fig. 2.3.1. Letu
(m ,t) be the displacement ofa ~a
P (m ) at time t with respect to the reference configuration, i.e. a ~a ~ u (m ,t) a ~a ~ ~ x (m ,t)-x (m ,t 0l a ~a a ~a + n AO Figure 2.3.1. Definition of s 0. (2. 3. 6)
If the rotations and displacements are small enough and the bound-aries are smooth enough, no penetratien occurs if
(2.3. 7) However, if the rotations and/or displacements are not small enough or the boundaries are not smooth enough, this inequality does not represent the impenetrability condition properly. An example of such a situation is sketched in fig. 2.3.2. Hence a more general approach
reference contiguration
current contiguration
..
..
..
Figure 2.3.2. Penetratien without vialation of (uA-u8l•nA0-s0iO
is needed. For this purpose it is assuaed that at each state t for all xerCA(t) a scalar quantity g = g(x,r
8(t)) can be defined, such
0 if
..
xer8(t) (2.3.8)The notation q
=
q(i,r8(t)) indicates that q depends on the vector
..
..
x and the shape of r
8(t). Therefore, g(x,r8(t)) should be considered as a functional rather than a function. From (2.3.8) it is seen that no pènetration occurs at state t if and only if
..
g(x,r
8(t)) i
o
..
v
xercA<t> (2.3.9)This inequality is a proper impenetrability condition, even in the case of large displacements and/or non-smooth boundaries.
The requireaents (2.3.8) imply a lot of treedom in the choice
.
..of the f~eld g(x,r
8(t)). In this thesis only one possibility is
exploited.
Consider a material point A with current position vector ierCA(t). In this point a unit vector; ;(~,t) is defined (see
tigure 2.3.3), such that the halfline
y
= i+À;, À€[0,•), intersects (or is tangent to) the boundary r8(t). Let n (nl1) be the number of
points of intersectien and let À
1, À2,
..
,Àn be the value of the parameter À in these points. The value g(x,r8(t)) can be defined such that
= { -min(À 1 ,À2 , ... , Àn) if i~i2
8
(t)+min(À1,À
2, ... , Àn) if xeQ8(t)
(2.3.10)
This field satisfies the requirements (2.3.8). It is easily seen that lg(x,r8(t) I is equal to the distance, measured in the direction vec
-tor;, from the point with position vector xerCA(t) to the boundary r
8(t) of body L8 at state t.
The unit vector ; plays a prominent role in the definition of g(x,r
8(t)). The actual choice of eisbasedon the following obser-vations. Firstly, not every search direction ; will yield one or more points of intersection. However, in that case x~Q
8
(t), hence, if..
necessary a negative value can be assigned to g(x,r
from a computational point of view it is advantageous if g{~,r
8
(t)) contains some information of what part ofrc
8{t) could be crossed inFigure 2.3.3. The halfline and points of intersection. a nearby future state by point A if ~is rather close to r
8{t) in the current state. Guided by these considerations ~ ~{~,t) is chosen as a given tunetion of the state, such that
a). ~is directed outwardof QA{t), i.e. ~.~A>O, if ~~Q
8
{t),.. ... .. +
-b). eis directed inward to QA{t), i.e. e•nA<O, if xeQ8{t),
0 0
In this thesis ~ is taken equal to the unit outward normal, respec-tively the unit inward normal, torAinsome reference configuration, which has tobechosen such that the requirement a)., respectively b}. is satisfied.
Summarizing, the contact conditions are
1). the law of action andreaction mustholdat current contact points, i.e.
(2.3.12)
2). the contact pressure must be non-positive, i.e.
(2.3.13)
•
3). a constitutive equation for ota must be given, 4). no penetration may occur, i.e.
(2.3.14)
5). if and only if q = 0 the contact pressure may be less than zero, i.e.
= 0 on (2.3.15)
2.4 Statement of the problem.
In this thesis only quasi-static isothermal deformation processes of LA and L
6 are studied. That is, inertia effects, stress relaxation, creep phenomena etc. are not taken into account.
r DB
Fiq. 2.4.1. Definition of the various quantities.
At the open bounded domain Q (t), occupied by body L (~=A,B) a
~ ~
qiven volume load
l
(r ,t) is applied. The boundary r (t) is split in~ ~~ ~
three disjoint parts
r
0a(t), fF~(t) and fC~(t) . At fF~(t) a qiven surface loadp
0(m ,t), which may be equal to zero, is applied, while~ ~~
it is assumed that alonq
re
(t) no external load is applied. Alonq~ .
r
0 (t) the confiquration field ~ (m ,t) is prescribed, i.e.x
(m ,t)~0 ~ ~~ ~ ~a
=x (m ,t). It is assumed that no riqid body motion of either of the
~ ~a
bodies is possible. Strictly speakinq this it is not necessary as lonq as the contact conditions quarantee that in the current state no riqid body motions are possible.
Finally, a constitutive equation for both ~ and ~t must be
~ a
qiven. Specifically, in this thesis primarily elasto-plastic material behaviour is considered.
(P).
Formally the problem under consideration is defined as
Find the configuration fields
x
(r ,t), a= A,B, and the stressa ~a
fields G , a ~t (a=A,B) and o that satisfy a n 1). the equilibrium equations
V•o
+l
= ~-a a ' G a Ge in a Q a (t),
2). the natural (= dynamicl boundary conditions
3). the essential (=kinematica!) boundary conditions
(2.4.1)
(2.4.2)
(2.4.3)
4). the law of action andreaction at the current contact sur-face rc(t) = rCA(tlnrc8(t)
and
5). the contact conditions (o n
(2.4.4)
(2.4.5)
(2.4.6)
..
6). both ota on rc(t) and Ga in Qa(t) satisfy a given constitu
2.5 A weak tormulation of the contact problem of two deformable bodies.
The principle of weighted residuals is a powerful tool to derive weak formulations for a large class of unconstrained continuum problems (Finlayson, 1976). A weak tormulation forms an excellentstarting point to generate approximate solutions. However, the weighted resi-duals technique is not directly applicable to constrained problèms such as contact problems ·, Some constrained problems can be handled by optimization techniques but this is not possible for many continuum problems of the type discussed in 2.4. Here an attempt is made to extend the principle of weighted residuals to handle a class of con-tact problems.
Let Y denote the space of kinematica! admissible contiguration a fields y a ... 3 ... {y :R -+R y (r ,t) a a a ~a -+0 x ( r , t) on
r
0 ( t)} , ( 2 . 5 . 1 ) a ~a aW the space of kinematica! admissible weight functions a
w
a{w
:R -+R3w
(r ,t) =0
on r0a(t)},
a a a ~a
and M the space of admissible contact pressures
(2.5.2)
(2.5.3)
The precise differentiability requirements of the elements of the above spaces form a rather delicate subject, because they heavily depend on the original problem (P). It is supposed that the elements of the above spaces satisfy all the continuity, differentiability and integrability requirements needed in the following analysis.
According to the principle of weighted residuals the equili-brium equations (2.4.1) are equivalent to
I Q a ... -t ... (V•~ +I )•W dQ = 0 a a a
...
v
w aew
a for a=A,B (2.5.4)~ ~ ~· ~ +
With (V•u }•w = -G :(Vw} + V•(G •w }, the application of Gauss's
a a a a a a
theorem, the symmetry of o and the natura! boundary conditions this can be written as .... c
I
UJ : (Vw } dQ Q a a..
..
L (x ,w } a a a - I r (a a a .~ )•W dr a a Ca V :ew
for a=A,B a a where the nonlinear functional L (~ .~ } is given bya a a
.. ..
L (x ,w } a a a1 ..
-+0 .. I •w dQ + I p •w dr Q a a r a a a Fa..
v
wew
a a 0 (2. 5 .5} (2.5.6)Let pa be the stress vector on r (t}, i.e. p = G .~ . On the
a a a a
actual contact surface re the law of action andreaction must hold, i.e. pA=-p8. Furthermore, on re the normals ~A and ~B must be
oppo-• . ~ .. • • • -+ .. ... .
s1te, 1.e. nA=-n
8. W1th the decompos1t1on p a =ot a na a +a n 1t follows
h .. .. .. .. . h' 1 . 2 5
t at otA=-otB and onA=-onB on re. Us1ng t 1s, the re at1ons ( . . 5} for a=A and a=B can be combined
r
[J G :(v~ }cdl2- L (~ .~ >la=A B Q a a a a a , a
(2. 5. 7}
Here, again, body LA is taken as the reference body (the contactor} and Ga•~a=O on rca\rc is used. The shear stress vector ~tA is deter -mined with the help of a constitutive equation. For numerical purposes it is rather inconvenient that (2.5.7} contains an integra-tion over the unknown true contact area. However this integration can be expanded to rCA by assigning toeach material point of rCA a
..
material point of rCB such that w
8 is defined at every point of rcA
..
· This can be done with the aid of the intersections of the halfline xA +Àe, ÀlO, with ree' see 2.3. Since GA.nA=O on rcA\rc, (2.5.7} can be written as[ [J ~ :(V~ )cdQ- L (~ .~ )]
a=A B Q a a a a a , a
(2.5.8)
The normal contact stress onA is considered as an unknowri ex-ternal force, which is to be determined such that the contact con-straints are satisfied. To stress the unknown character of onA a new variable, the contact pressure p, is introduced where p=onA on rCA"
Of course p€M since p is always non-positive. The normal ~A in
(2.5.8) may be replaced by -~B. On the one hand this relates the
cur-rent tormulation directly to the Lagrange multiplier tormulation (see
2.6) and on the other hand this turns out to have some computational
advantages. Hence, (2.5.8) can be rewritten, yielding
[ [J ~ :(V~ )cdQ- L (~ .~ )]
a=A B Q a a a a a , a
(2.5 .9)
The contact conditions must be taken into account in an additional
relationship. The impenetrability condition g i 0 on rCA can be
writ-ten as
(2. 5.10)
The compatibility condition pg=O (see 2.3.15) can be added to the integrand which results in the inequality
f (q-p)g dr l 0 V q€M
rCA
(2.5.11)
Now, the earlier mentioned extension of the principle of weigh-ted residuals can be staweigh-ted as follows
..
..
Tbeorem 2.5.1 A solution (xA,x
8,p)€YAxY8xM of the problem
(PW). [ [J 111 :(V~ )cdQ- L (x.~)]= a=A,B Q a a a a a a I (q-p)q dr
2
o
v
qeM rCA (2.5.12) (2.5.13)..
where Cl and ot a a where L (x a a a .~ )(a=A,B) satisfy the qiven constitutive relations and is qiven by (2.5.6), is a weak solution of problem
(P) as well.
Proof. First it is sbown that a solution of (PW) satisfies the con-tact conditions (2.4.6). Because peM it follows from q=p+z that q€M for each zeM. Therefore (2.5.13) can be written as
J zq dr
2
o v
zeM rCA(2. 5 .14)
hence qiO on rCA. By takinq subsequently q=O and q=2p in (2.5.13), it can easily be shown that
(2.5.15)
and because both PiO and qiO this can only hold if pq=O on rCA. The complementary condition pq=O toqether with PiO and qiO im-plies that the contact pressure can only be neqative if q=O, i.e. only on the actual contact surface re. This result, combined with the substitution of 111 :(V~ )c = -(V•o )•~ +V•(o .~) in (2.5.12) and tbe
a a a a a a
application of Gauss's tbeorem, yields
1
~...
... ..
...
[ [J (- -V•e~ )•w dQ + J (e~ •n -p )•w dr] +
a=A B Q a a a r a a a a
(2.5.16)
This results in the equilibrium equation, the dynamic boundary condi-tions on rFa and w
8
•~8
=~ on re8\re. From geometrical arguments it is..
..
known that nA=-n
8 on re. Therefore the right hand side of (2.5.16) yields
..
..
..
wA•nA = 0tA+pnA on re (2.5.17)
..
..
..
ws•ns = -otA+pnB on re (2.5.18)
Hence the law of action andreaction applies on re, i.e.
(2.5.18)
Since on reA\re the distance functional g must be negative, and pg=O, the contact pressure p must be zero on reA\re. Furthermore, the constitutive relation for ~tA must be such that ~tA=~ if p=O. Therefore wA.~A=~ on reA\re, and this completes the proof.
2.6 The Laqranqe multiplier metbod in case of frictionleas finite elasticity contact problems.
The Lagrange multiplier method is a widely used technique to solve constrained optimization problems (Luenberger, 1969). However, only a limited class of contact problems can be written as an optimization problem. This, for instance, is the case if the bodies behave hyper-elastic, the applied forces are conservative and no friction occurs. It is useful to study the Lagrange multiplier method because in the discretized situation, conditions for the uniqueness and existence of the Lagrange multiplier can be given. It is shown that the Lagrange
multiplier method can beregardedas a special case of problem (PW), where the Lagrange multiplier is to be considered as the contact pressure field. Therefore it is reasonable to apply these conditions for uniqueness and existence to (PW) as wel!. This does not imply that the contact pressure field in (PW) actually exists.
For conservative loads a potential U(y l can be defined such
a
that the Gateaux-derivative of U at ~ in the direction ~ is given
a a by
..
..
(0 U (X ) ;w> =
a a a a lim 8+0 U (~ +8~ )-U (~ ) a a a a a 8..
..
= L (x ,w l a a awhere L (~ .~ l is defined previously in (2.5.6): a a a L (~ I~ ) a a a ~ .. +0 .. I r •w dQ + I p •w dr Q a a r a a a Fa (2. 6.1) (2.6.2)
If the bodies behave hyperelastic there exists (at least locally) a stress free configuration. Let ~ be the Green-Lagrange strain
ten-a
sor with respect to this so-called natura! reference configuration. Then there exists a convex stored energy function ~ with respect to
a
the reference contiguration such that the second Piola-Kirchhoff stress tensor $ is given by
a $ :4! a lim 9+0 ~ (~ +94!)-~ (E l a 11 a a 8 YC I 4Jc = 4!; $ a (2. 6. 3)
For hyperelastic bodies L under conservative loads, the potential
a
.
..
energy funct1onal F :y +R can be written as
a a
..
F (y ) a a 1 • J -- ~ (E ) dQ - U(y
)
Q 6a a a a a a (2. 6. 4)where 6 =det(f ), a a i.e. 6 is the volume a change factor with respect to the natura! reference configuration.
The space K of kinematica! admissible contiguration fields is defined as
So, each element of K is a combination of the two contiguration fields yA€YA and y
9eY8, such that at no point of rCA(t) the impene-trability constraint is violated.
If it can be shown that a (local) minimizer (xA,x
8)€K of the
.
.
..
...
.
.
total potentlal energy funct1onal FA(yA)+F
9(y9) 1s a solut1on of the original problem (P) without friction, then the Lagrange multiplier method can be applied. The next theorem establishes this.
Theorem 2.6.1. Let the potential energy functional of body L be a. given by (2.6.4) and let the boundaries rCA(t) and rc8(t) have no corners. Then, a local minimizer of FA+F
9 is a solution of the origi-nal problem (P) without friction. Hence for hyperelastic bodies under conservative loads, the original problem is equivalent with
..
..
(PM). Find a pair (xA,x9)eK such that
( 2. 6. 6)
Proof. The proof given here is analogous to the proof provided by Ciariet and Necas (1985), who studied the combination of a rigid body and a deformable body. Consider
y €Y as a pertubation of x of the
a. a. a. form...
...
x +ew a. a.
...
w a. ew a. (2.6.7)0 • ~ ~ ~ ~ • •
There exist f1elds wAewA and w
8ew8 such that (xA+wA,x8+w8)eK, wh1ch implies that (yA,y
9)eK for each e€(0,1). Since (2.6.6) holds for
..
..
.
...
..
every (yA,y8)eK, the Taylor-ser1es expansion of FA(yA)+F9(y9) gives
[ [F
(y
>-F(x )]
a.=A,B a. a. a. awhere <D F (~ );E~ > denotes the Gateaux derivative
IX IX a ( l
the direction
w
.
This derivative is linear in E~ •of F at ~ in
a a
IX a In appendix A it
is shown that (2.6.8) results in
+0 .. . 2 .. .. ..
- f p •EW dr) + O(E ) l 0 Y (Wal (yA,yB)€K) (2.6.9)
r a ( l
FIX
and using Green's theorem this can be written as
.,
..
..
[ [f (-I -V•o )•EW dQ + f
~X=A,B Q a a a
r
.. +0 ..
<~ •n -p )•Ew dr +
a a a a
a Fa
From this inequality it is easily seen that
and
...
~ •n IX a ~ in Q a +0 p ( l on rFaHence, the following inequality results
It will be
...
= ~ 0 ta on on reB ther
r
Ew
a·~
a•n
a dr + o(e2> a=A,B reashown that this implies ~
•n
a al 0
= ~ re fora= A,B. First, body LA is inequality ( 2. 6. 13) reduces to (2.6.10) ( 2. 6. 11) (2.6.12) ( 2. 6. 13) on re \re and a
<
0, a na -considered. W~th.
w...
=ó
8 (2.6.14)Thus,
..
~A.~A = Ö on reA\re. Using this result and the decomposition..
..
~A·nA = atA+anAnA, the inequality (2.6.14) becomes
(2.6.16)
..
..
..
Step 2. Let tA be a unit tangent vector to re and take wA = ~AtAewA.
..
..
Then g(xA+ewA,r8(t))=O on re for all e€(0,1), and (2.6.16) results in
..
..
ö
for every ~A and every tangent vector tA. Thus atA = on re and the inequality (2.6.16) becomes:
..
..
.
..
..
step 3. Take wA = ~nAewA w1th ~<O on re. Then g(xA+ewA,r8(t))iO on re
for every e€(0,1), and (2.6.18) yields the requirement
(2.6.19)
for each ~ with ~<O on re. Thus anA<O on re. Body t
8 can be treated likewise, resulting in ~
8
·~8
=Ö on re8\re and an8iO,~ts=Ö
on re.Finally it is shown that the law of action and reaction holds on
..
..
ö
re. Because atA = atB = on reit is sufficient to prove that anB
-anA on r C"
..
..
Using the preceding results and takinq w8=wA on re the inequality (2.6.13) becomes
{2.6.20)
for every ~AewA. Hence anB = -anA" This completes the proof.
Solving problem {PM) is not straightforward. Moreover it is not suitable to derive practical computational techniques. This is mainly caused by the fact that it is very difficult to define contiguration
..
..
fields yA and y
8 such that the impenetrability condition giO is satisfied a priori.
A very common technique to release this eenstraint is to apply the Lagrange multiplier rule. Let the nonlinear functional
{2. 6. 21)
...
...
. ..
Let {xA,xB)€K be a local m1n1m1zer of the total potential
ener-.
..
...
.
gy funct1onal F{yA)+F
8{y8). Then, accord1ng to the Lagrange multi-plier rule {Luenberger, 1969), there may exist a Lagrange multiplier
..
...
...
..
field p€M such that L(yA,y
8,p) is stationary at (xA,x8,p) while the complementary condition pg=O is satisfied on rCA" With this rule, the tormulation (PL), as stated in the next theorem, is derived. It can beseen as a special case of problem {PW) of theorem 2.5.1.
Theorem 2.6.2. A salution of the problem
(PL). Find a triple (~A'~
8
,p)€YAxY8
xM such thatr
[f o :(V~ lcdQ-L <~ .~ lla=A,B Q a a a a a
a
(2.6.22)
qiO, pg=O on rCA {2.6.23)
is a stationary point of the Lagrangian (2.6.21) and is also a weak salution of the original problem (P) in the frictionless situation. The multiplier field p is the current contact pressure.
~ ~
.
~ ~Proof. In a stationary point (xA,x8,p) of the Lagrang1an L(yA,y 8,p) the Gateaux derivative of L(.) with respect to the first two argu-ments is equal to zero, i.e.
[ [<D F (x);~ >-J p<D g;~ >dr] = 0
=A 8 ~ a a a r a a
a , ~
(2.6.24)
With the results of the appendices A and B, equation (2.6.21) can easily be derived. Following a similar argument as in theorem 2.5.1, the multiplier p can be identified as the current contact pressure.
~ ~
That (xA,x8,p) is a weak solution of the frictionless problem (P) follows immediately from theorem 2.5.1. This completes the proof.
2.7 Other methods.
It is well-known that the Lagrange multiplier metbod has some draw-backs of computational nature. First of all the number of unknowns is increased by adding the Lagrange multiplier to the set of unknowns.
Furthermore, the structure of the linearized system of equations has some awkward characteristics. There are several ways to overcome the above problems. One can either try to find another formulation with-out these problems, or one can try to solve them on a computational level. In this thesis the latter has been chosen, but to give a more complete treatment of the contact problem and to justify our choice two other formulations will be discussed.
In 1941 Courant introduced the exterior penalty metbod to solve constrained boundary value problems. With respect to geometrical linear contact problems, this metbod has been studied extensively by Oden, Kikuchi and Song (1980). There are two approaches to introduce the penalty function method. The original approach adds a penalty functional to the potential energy functional and considers the sta-tionary conditions of the new functional. A more recent approach, as
introduced by Bercovier (1979), is by perturbing the Lagrange multi-plier formulation. This approach can easily be applied to the weak tormulation (PW).
Consider the inequality (2.5.13), i.e. J (q-p)g dr l o v qeM
rCA
(2.7.1)
This inequality is equivalent to the contact conditions pg=O and giO on rcA· In the perturbed Lagrange multiplier tormulation the impene-trability condition giO is weakened by replacing it by g+epeiO with e>O. The penalty factor E is chosen such that ep is smal! as
com-e
pared to some characteristic lengthof the contact bodies. The multi-plier pe is the new unknown in stead of p. The problem (PW) is then modified to .... c . . . . [ [J ~ :(Vw) dQ-L (x ,w )] = a=A B Q a a a a a , a (2. 7 .2)
J (q-p )(g+ep) drlO v qeM
r E E
CA
(2.7.3)
The penalty function tormulation fellows from (PW ) after eli
-E
minatien of p . In the usual manner (see 2.5) the inequality (2.7.3)
E yields p (g+ep ) E E (g+ep ) < E - 0 0 on rCA on rcA So, from (2.7.4) it fellows that if
(2.7.5) it fellows that if g=O then g+ = max{O,g). Substitution of this penalty tormulation 1 p <O then p =--g, E E E p <0. Hence, p = e- e result in (2.7.2) (2.7.4) ( 2. 7. 5) while from 1 + -ëg , where yields the
~~ c ~ ~
[ [J m :(VW) dQ-L (x ,w )]
a=A,B Q a a a a a a
(2.7.6)
A serious problem with this tormulation is that the penalty factor E
must be taken sufficiently small as to guarantee a good approximation of the eenstraint giO. Especially in non-linear problems this leads to a rather poor condition of the tormulation which makes the problem very difficult to solve.
To evereome some of the computational difficulties of both the Lagrange multiplier metbod and the penalty tunetion methad the aug-mented Lagrange metbod was introduced by Powell (1969) and Hestenes
(1969) and investigated in detail by Rockafeller (1973) and Bertsekas (1982). Examples of applications of the metbod to boundary value pro-blems can be found in Fortin and Glowinski (1983). Basically, the methad adds a penalty functional to the Lagrangian in stead of to the potential energy functional. This has several advantages. One of them is that the penalty factor does not have to be chosen as small as in the classica! penalty function formulation. It can be taken suffi-ciently large to maintain the original condition of the formulation.
Further the contact conditions can be satisfied exactly, at least in the continuum case. So, let the augmented Lagrangian be defined as:
(2.7.7)
Following an analogous argument as in 2.6 the augmented Lagrangian tormulation can be stated as
r
[J m :(V~ )cdQ- L (x.~)]a=A,B Q a a a a a a
I
(q-p)g dr l ov
qeMr~
(2.7.9)
The problem with this method is that it is not known a priori what value should be given to t as to maintain a good condition and still
Chapter 3 A constitutive eguation for a class of frictional phenomena.
3.1 Introduction.
In many applications, especially in problems where smal! displace-ments and deformations occur, it is satisfactory to use the classica! Coulomb law to describe the frictional behaviour between contact bodies. However, in metal forming processes where large displace-ments, rotations, deformations and high contact pressures occur, this model leads to erroneous results. Many experiments have shown that the coefficient of friction is not, as is usually assumed, a constant and that in many cases the shear stress is not proportional to the contact pressure. A first attempt to give a better description of frictional aspects in metal forming problems was the introduetion of the so-called constant friction factor model. In this model the magnitude of the shear stress vector is taken proportional to a relevant yield stress. Still, the results obtained by this model are quite unsatisfactory, both from a practical and a theoretica! point of view. Therefore a more general approach is desired.
This chapter attempts to give a phenomenological description of a fairly general class of frictional phenomena. Following the work of Fredricsson (1976), Petersson (1977), and Michalowski and Mroz
(1978), the description is basedon the observation that there is a close resemblance between frictional and elasto-plastic behaviour. An essential feature of the present model, distinguishing it from the models proposed by these authors, is that it allows for large relati-ve motions of the contact bodies. This has a tremendous impact on the mathematica! description of the considered frictional phenomena.
The derivation in this chapter is to a great extend analogous to the derivation of constitutive equations for large strain elasto-plastic material behaviour. The key elements of the present model are
- the use of objective rates,
- the recognition of reversible and irreversible aspects of slip (like the decomposition of the deformation rate in an elastic and a plastic part),
- the introduetion of a slip tunetion and a slip criterion (analogous to the (von Mises) yield tunetion and the yield criterion),
- the proposition of a quasi associated slip rule (which is an analogon of the associated flow rule),
- the incorporation of history dependent quantities (similar to the hardening models used in elasto-plasticity).
The classica! Coulomb model as well as the constant friction factor model can easily be shown to be special cases of the present model.
First, a geometrical description of the contact boundaries is given and a number of kinematica! quantities are introduced. All considerations are limited to contact points which remain in contact at least temporarily, so points which come in contact or are loosing contact are not studied. Next stresses are considered. The shear stress vector always has to satisfy the so-called orthogonality rule as well as the law of action and reaction. These two conditions are derived in a rate-type form. This is necessary because, due to the inclusion of history dependency, a rate-type tormulation is required.
Thereafter, the consequences of the principle of objectivity are studied and it is shown that the material rate of the shear stress vector is not suitable for direct use in a constitutive equation. A class of objective rates is derived which is applicable in formula-ting a constitutive equation.
To characterize the irreversible aspect of slip a slip function,
a slip criterion and a quasi associated slip rule are introduced. A
constitutive equation for irreversible slip can be derived after a specification of the history parameters.
3.2 Some kinematic guantities.
Some additional kinematic quantities, specific for this chapter, have to be introduced, while some of the previously defined contact conditions are rewritten in a form more suitable for current purpo-ses.
The velocity
v v
(m ,t) ofa a ~a P a (m ) ~a is equal to the B -rate of a
0 0 . . . . . .
the pos1t1on vector x =x (m ,t).
a a ~a Formally, the B -rate of a quan-a tity (scalar, vector or tensor) w
a w a (a ~ ,t) is given by
w
a 1 lim •t[w (m ,t+llt)-w (m ,t)] llt ...o
u a ~a a ~a ( 3 0 2 0 1)where the dot in combination with the
...
the quantity. The current velocity v a posed into the normal velocity
v
insuffix a denotes the B -rate of a
of P (m ) is additively decom-a ~a
the direction of the unit
... ... na
outward normal n = n (m ,t) on r (t) in P (m ) and the tangential
a a ~a a a ~a
...
velocity vta..
...
...
...
V = V n a; V V •n na na na a a (3.2.2)...
...
...
([-n
rt
)•v
vta V -v = a na a a · aLet
v
be the gradient operator in the tangent plane of r (t) ina a
P (m ), i.e. it is the projection of the usual three dimensional a ~a
gradient operator onto the tangent plane. This operator is discussed in more detail in appendix e. Within re the gradient operator with respect torA, i.e. VA, cannot be distinguished from the gradient
...
...
operator with respect to r8, i.e. v
8. So, for all x ere the expres-sion v w
(~
,t) can be written as vw(~
,t), where~is
the gradienta a a a a
operator in the tangent plane of re(t) at a current contact point. All further observations are restricted to current contact points .
...
Because LA and L8 are impenetrable the normal veloeities vnA and
...
vnB of current contact points have to satisfy the condition
(3.2.3)
...
The slip velocity v
8A between the current contact points PA(~A) and
P
8
(~8
) is defined as the tangential velocity of point P8
(~8
) with respect to point PA(~A)Here and in (3.2.3) LA is considered as the reference body. This choice is arbitrary since L
8 could be taken as the reference body as well.
It is easily seen that a current contact point PA(~Al remains in
.
.
.
.
~ ~ ~contact, at least temporar1ly, w1th r
8 1f and only 1f (v8-vA)•nA 0. In this case, the relation for the slip velocity ~BA becomes ~BA
..
..
vB-vA.
At current contact points PA(~A) and P
8
(~8
) the law of action and reaction applies. As a consequence the normal stress veetors and the shear stress veetors in these points must satisfy(3.2.5)
..
..
It is noted that ota and na are always orthogonal. Taking the B -rate of ~
•n
=o
yieldsa ta a
and because the B -rate of ~ is related to ~ by
a a a
..
n a....
..
-(Vv )•n a a..
it follows that ota always has to satisfy
~ • ~· c ~
n a •[at a -(Vv ) a •o ] ta 0
(3.2.7)
(3.2.8)
If PA(~A) remains in contact, at least temporarily, with r 8 and if P
8
(~8
) remains in contact, at least temporarily, with rA, it can... ... ::t ... ..
be shown from otA+otB = u that the BA-rate otA of otA and the
a
8-rate..
..
otB of otB are related by (see appendix Dl
3.3 Objectiye rates of the shear stress yector.
Each imaginary motion of LA and L8, that can be derived from the actual motion by a rigid rotation ~(t) and a rigid translation u(t) of both LA and L
8, is called an objective equivalent motion of LA and L8. Here 0 is a rotation tensor, i.e.
det(Ol = (3;3.1)
Let w be a quantity (scalar, vector or tensor), associated with the
a
actual motion. The cortesponding quantity, associated with an
objec-*
tive equivalent the velocity of
motion is denoted by w . For the position
a
a material point P (m ) of r (a=A,B) this
a ~a a vector and implies
.. *
x (m , t) = a ~a.. *
v (m ,t) = a ~a..
..
u(t)+O(t)•x (m ,t) a ~a (3.3.2) ~.
..
..
u(t)+~(t)•x (m ,tl+;(t)•v (m ,t) a ~a a ~aThe unit outward normal on r in P
<• )
and the gradient operator ina a ~a
the tangent plane of r~ in P (m ) will transform according to .... a ~a
.. *
..
V = II!•V
Combining these results with (3.3.2) it is easily shown that .... c .. .. ·c
;e(Vv )•~ +~•([-n n )•~
a a a
(3.3.3)
(3.3.4)
A vectorial quantity ~ and a tensorial quantity W are objective if
a a
they transfora according to
(3.3.5)
for every u and
~.
They are invariant if~*
=~
andw*
= W for a a.. a.... aevery
u
ando.
Hence,n
is objective whilex
I V and vv are not.a a a a
The physical principle of objectivity states that the stress vector
..
..
..
...
..
a q)•o ·
na na' O•o ta
..
(3.3.6)A problem in formulating rate-type constitutive equations for friction is that the B -rate of an objective vectorial quantity is
~
not objective. Taking ota as an example it follows that
(3.3.7)
Similar problems are encountered in formulating rate-type constitu-tive equations tor large deformation problems (Nagtegaal and de Jong (1982), Lee et. al. (1983), Moss (1984)). To overcome this problem a similar procedure is used as in van
.
..
..
S1nce ata and n are orthogonal the a
.. ..
..
4)•([-n n )•ot . Using ( 3. 3. 4) it is a a a
-+* ... c ... ot a -(V v ) ex •ot a
Wijngaarden and Veldpaus ( 1986).
..
term O•ota in (3.3.7) is equal to possible to eliminate 4), yielding
(3. 3. 8)
Therefore ... ot - (Vv ) •ot .... c .. V a ex a
..
is objective, while ata is not. Furthermore, each rate
..
otex' dehned by.
~
0 :::
ta (3.3.9)
is objective if and only if H is objective. The tensor H is intro-a ex duced such that different objective rates can be introduced analogous to those used in constitutive equations for elasto-plastic material behaviour. In this thesis only tensors 8 and correspondinq objective
V a
..
.
.
.
rates ota w1ll be cons1dered that sat1sfy some additional
requirements. Firstly, H may only depend on kinematica! quantities ex
of body L . Secondly, the chosen tensor H must result in an objec-a a
.
~.
.
..
.
t1ve rate ota wh1ch 1s orthogonal to na. Us1ng (3.2.8) and (3.3.9)
..
..
this requires that ota and Ha•na are always orthogonal, i.e.
.. .. ;t
n *(H •n ) =u
V where
*
denotes the vector product. Thirdly,~ if the motion of LA and t8 at that time is
..
.
ata at t1me t has to be a rigid body rotation.
~ .... c .. ~ c ..
In that case at a -(Vv) •at a = u and hence H •at must be equal a a As a consequence, for any rigid body motion of both bodies H
a be of the form
..
..
11 n h a a a to ~ should (3,3.11)where
h
is a vector that, because of previous requirements, must be adetermined by kinematica! quantities of L . An 11 that satisfies all
a a
the above requirements is the null-tensor.
In the next section !ta will be used to formulate a constitutive equation and it is necessary to reformulate the requirements (3.2.8)
~
and (3.2.9) in terms of ata" The rate-type orthogonality condition (3.2.8) can be written as
.. ~ c ..
n •(at a a a +11 •at ) a 0 (3.3.12) while the rate-type law of action andreaction (3.2.9) results in
(3.3.13)
It is easily shown that this equation is invariant for interchanging the indices A and B.
3.4 Reversible and irreversible behayiour.
Most classica! friction models only take account of irreversible behaviour. However, experiments of Fredricsson (1976) show that fric-tion also has a reversible part. A possible explanafric-tion of this phenomenon is that the asperities in the contact surface behave elastically before essential irreversible slip will occur. Further-more, there may be a very thin film of some lubricant between the
contact surfaces and this lubricant can behave elastically for small relative displacements of those surfaces. Taking into account the reversible part of friction has some numerical advantages (Oden and Pires, 1983).
The modelling of frictional behaviour will be illustrated for the case of dry friction where the true contact takes place at the asperities. First of all it is assumed that there is a one to one correspondence between the (physical) asperities and the
(mathematica! tietion of) material points of the boundary r . The a
velocity of an asperity with respect to the associated material point
..
P (m ) is taken as a measure for the rate of the shear stress ota at
a ~a. . . . .. .. .. .
P (m ). Th1s veloc1ty 1s equal to vh -v, where vh 1s the velocity
a ~a a .. a .. a ..
of the asperity . It is assumed that vh -v is orthogonal to n , i.e.
a a a
..
..
..
(vh -v )•n a a a = 0. S1nce vh -v 1s ob]ect1ve an ob]ect1ve rate for the
.
...
..
.
.
.
.
.
a ashear stress has to be used in the constitutive equation. The postu-lated constitutive equation is
for a=A,B (3. 4. 1)
where the so-called flexibility tensor IM is symmetrical, positive a
As a consequence of the assumptions (vha-definite and objective.
V
...
...
...
..
v )•n a a 0 and ot •n a a 0, the normal ~ must be an eigenvector of a
M . The corresponding
a eigenvalue m must be positive as a consequence a
of the properties of IM a
..
..
IM •n a a = m n · a a' m >0
a (3.4.2)
It is difficult to determine the tensor M for each body separately,
a
but often it is possible to give an estimate for the sum IMA~B. In these situations MAand
'Ka
will be taken equal. In the isotropie caseMA~B
can be written as t-1[, where t is some stiffness parameter.To model reversible slip it is assumed that at current contact points PA(~A) and P
8
(~8
) the relative velocity between the asperity associated with PA(~A) and the asperity associated with P8
(~8
) is1 . .. .. ::t
momentary equa to zero, 1.e. vhA-vhB =u. With the use of (3.4.1) in V
the modified form vh
